Answer

Verified

382.5k+ views

**Hint:**In elementary algebra, root rationalisation is a process by which radicals in the denominator or numerator of an algebraic fraction are eliminated. To rationalize a term we need to multiply it with a term such that the square root, cube root etc is eliminated. We multiply with the same term in denominator and numerator to maintain the original value of the fraction which is to be rationalized.

**Complete step by step answer:**

Here we have the fraction \[\sqrt{\dfrac{\cos x}{\sin x}}\] where we need to rationalize the numerator.

In fraction we have two term numerator and denominator. The term above the line is called numerator and the term below the line is called denominator. For example in the fraction

\[\dfrac{2}{3}\], the above term that is 2 is numerator and the below term, 3, is the denominator.

Thus we can clearly see in the term \[\sqrt{\dfrac{\cos x}{\sin x}}\] the numerator is \[\sqrt{\cos x}\] and denominator is \[\sqrt{\sin x}\]

Now to rationalize the numerator we need to multiply it with a term such that the square root is removed. As we know multiplying two same square root terms result in same term , we will multiply both denominator and numerator by \[\sqrt{\cos x}\]

\[\Rightarrow \sqrt{\dfrac{\cos x}{\sin x}}\left( \dfrac{\sqrt{\cos x}}{\sqrt{\cos x}} \right)\]

(We know that \[\dfrac{\sqrt{\cos x}}{\sqrt{\cos x}}=1\] and multiplying anything with 1 will not alter its value)

\[\Rightarrow \sqrt{\dfrac{\cos x(\cos x)}{\sin x(\cos x)}}\]

\[\Rightarrow \sqrt{\dfrac{{{(\cos x)}^{2}}}{\sin x(\cos x)}}\]

\[\Rightarrow \dfrac{\sqrt{{{(\cos x)}^{2}}}}{\sqrt{\sin x(\cos x)}}\] , (by using the property \[\Rightarrow \sqrt{\dfrac{a}{b}}\Leftrightarrow \dfrac{\sqrt{a}}{\sqrt{b}}\] )

\[\Rightarrow \dfrac{(\cos x)}{\sqrt{\sin x\cos x}}\] , (by using the property \[\Rightarrow \sqrt{{{a}^{2}}}=a\] )

**Hence the rationalized term required is \[\dfrac{(\cos x)}{\sqrt{\sin x\cos x}}\] , (only numerator rationalized).**

**Note:**The rationalization of a fraction or a term is done to simplify it and to find out the actual value of the term in a more simple form. Generally the rationalization is done for the denominator, unlike in the above question, to make the denominator the same/rational in case there are multiple terms in an expression.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths